In a small bowl, whisk together mayonnaise and mustard - season with salt and fresh ground black pepper. Spread half of this mixture over 4 slices of bread. On the same four slices, layer each with 2 ounces ham and about 1 1/2 ounces cheese.

Evenly spread the rest of the mayonnaise mixture between the remaining flour slices of bread. Place bread, mustard side down, on top of the sandwiches.

In a shallow dish, whisk together egg, egg whites and milk - season lightly with salt and fresh ground black pepper. Dip both sides of each sandwich into the egg mixture.

In a large skillet or griddle, melt butter over medium. Transfer sandwiches into the pan and cook until each side is golden and the cheese has melted, about 3 to 4 minutes per side.